1. Standard form Operations The Cartesian Plane Modulus and Arguments
COMPLEX NUMBERS
Standard form
Operations
The Cartesian Plane
Modulus and Arguments
nhaa/imk/sem /eqt101/rk12/32

2. Classification of Numbers
INTEGERS
(Z)
COMPLEX NUMBERS
(C)
REAL NUMBERS
(R)
RATIONAL NUMBERS
(Q)
IRRATIONAL NUMBERS
( )
WHOLE NUMBERS
(W)
NATURAL NUMBERS
(N)
nhaa/imk/sem /eqt101/rk12/32

3. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
In real life, problems usually involve Real Numbers(R). Imaginary number: If we combined Real number and imaginary number:
A number that cannot be solved.
nhaa/imk/sem /eqt101/rk12/32

4. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Why do we need to study complex numbers, C ?
Many applications especially in engineering:
Electrical engineering, Quantum Mechanics and so on.
Allow us to solve any polynomial equation, such as:
nhaa/imk/sem /eqt101/rk12/32

5. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
To solve algebraic equations that don’t have the real solutions
Since, the imaginary number is then
Real solution
No real solution
nhaa/imk/sem /eqt101/rk12/32

6. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Simplifying a complex number:
Since we know that
To simplify a higher order of imaginary number:
nhaa/imk/sem /eqt101/rk12/32

7. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Try to simplify
Solution
nhaa/imk/sem /eqt101/rk12/32

8. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Simplify
(a)
(b)
(c)
nhaa/imk/sem /eqt101/rk12/32

9. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Simplify
(a)
(b)
(c)
Answer:
nhaa/imk/sem /eqt101/rk12/32

10. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Definition 1.1
If z is a complex number, then the standard equation of Complex numbers, C denoted by:
where
a – Real part of z (Re z)
b – Imaginary part of z (Im z)
nhaa/imk/sem /eqt101/rk12/32

11. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Example: Express in the standard form, z: Answer: a) b)
nhaa/imk/sem /eqt101/rk12/32

12. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Definition 1.2
2 complex numbers, z1 and z2 are said to be equal if and only if they have the same real and imaginary parts:
If and only if a = c and b = d
nhaa/imk/sem /eqt101/rk12/32

13. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Example : Find x and y if z1 = z2:
nhaa/imk/sem /eqt101/rk12/32

14. nhaa/imk/sem120162017/eqt101/rk12/32
Introduction
Example : Find x and y if z1 = z2: Answer: a) b)
nhaa/imk/sem /eqt101/rk12/32

15. Operations of Complex Numbers
Definition 1.3 If z1 = a + bi and z2 = c + di, then:
nhaa/imk/sem /eqt101/rk12/32

16. Operations of Complex Numbers
Example Given and find: Answer: a) b) c)
nhaa/imk/sem /eqt101/rk12/32

17. Operations of Complex Numbers
Definition 1.4
The conjugate of z = a + bi can be defined as:
the conjugate of a complex number changes the sign of the imaginary part only!!!
obtained geometrically by reflecting point z on the real axis
Im(z)
Re(z) 3 -3 2
z(2,3)
nhaa/imk/sem /eqt101/rk12/32

18. Operations of Complex Numbers
Example : Find the conjugate of: Answer: a) b) c) d)
nhaa/imk/sem /eqt101/rk12/32

19. The Properties of Conjugate Complex Numbers
nhaa/imk/sem /eqt101/rk12/32

20. Operations of Complex Numbers
Definition 1.5 (Division of Complex Numbers) If z1 = a + bi and z2 = c + di then:
Multiply with the conjugate of denominator
nhaa/imk/sem /eqt101/rk12/32

21. Operations of Complex Numbers
Example: Simplify and write in standard form, z: Answer: a) b) c)
nhaa/imk/sem /eqt101/rk12/32

22. The Complex Plane/ Cartesian Plane/ Argand Diagram
The complex number z = a + bi is plotted as a point with coordinates z(a,b). Re (z) x – axis Im (z) y – axis
Re(z)
O(0,0) a b
Im(z)
z(a,b)
nhaa/imk/sem /eqt101/rk12/32

23. The Complex Plane/ Cartesian Plane/ Argand Diagram
Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by *Distance from the origin to z(a,b). r
O(0,0)
Re(z) a b
Im(z)
z(a,b)
nhaa/imk/sem /eqt101/rk12/32

24. The Complex Plane/ Cartesian Plane/ Argand Diagram
Definition 1.7 The argument of the complex number z = a + bi is defined as arg(z) is not unique. Therefore it can also be written as:
1st QUADRANT
2nd QUADRANT
4th QUADRANT
3rd QUADRANT
nhaa/imk/sem /eqt101/rk12/32

25. The Complex Plane/ Cartesian Plane/ Argand Diagram
Example: Find the modulus and the argument of z: Answer: a) b) c) d) e)
nhaa/imk/sem /eqt101/rk12/32

26. The Complex Plane/ Cartesian Plane/ Argand Diagram
The Properties of Modulus
nhaa/imk/sem /eqt101/rk12/32

27. Polar form Exponential Form De Moivre’s Theorem Finding Roots
COMPLEX NUMBERS
Polar form
Exponential Form
De Moivre’s Theorem
Finding Roots
nhaa/imk/sem /eqt101/rk12/32

28. The Polar Form of Complex Numbers
(a,b) r
Re(z)
Im(z)
Based on Figure 1:
Applying the Pythagorean trigonometric identity,
Therefore,
(1)
nhaa/imk/sem /eqt101/rk12/32

29. The Polar Form of Complex Numbers
The standard form of complex numbers is given by: Definition: Then the polar form is defined by:
nhaa/imk/sem /eqt101/rk12/32

30. The Polar Form of Complex Numbers
Example: Represent the following complex numbers in polar form: a) b) c) Answer :
nhaa/imk/sem /eqt101/rk12/32

31. The Polar Form of Complex Numbers
Example: Express the following in the standard form complex numbers: a) b) c) Answer :
nhaa/imk/sem /eqt101/rk12/32

32. The Polar Form of Complex Numbers
Theorem 1: If z1 and z2 are complex numbers in polar form where, Then, a) b)
nhaa/imk/sem /eqt101/rk12/32

33. The Polar Form of Complex Numbers
Example: Find and , if: a) b) Answer: a) b)
nhaa/imk/sem /eqt101/rk12/32

34. The Exponential Form of Complex Numbers
Euler’s formula state that for any real number Where is the exponential function, i is the imaginary unit, sine and cosines are trigonometric function and arg (z) = is in radians. Definition: The exponential form of a complex numbers can be defined as:
nhaa/imk/sem /eqt101/rk12/32

35. The Exponential Form of Complex Numbers
Example: Represent the following complex numbers in exponential form: a) b) c) Answer :
nhaa/imk/sem /eqt101/rk12/32

36. The Exponential Form of Complex Numbers
Example: Express the following in the standard form complex numbers: a) b) Answer :
nhaa/imk/sem /eqt101/rk12/32

37. The Exponential Form of Complex Numbers
Theorem 2: If z1 and z2 are complex numbers in exponential form where, Then, a) b)
nhaa/imk/sem /eqt101/rk12/32

38. The Exponential Form of Complex Numbers
Example: Find and , if: a) b) Answer: a) b)
nhaa/imk/sem /eqt101/rk12/32

39. nhaa/imk/sem120162017/eqt101/rk12/32
De Moivre’s Theorem
Let z1 and z2 be complex numbers where Then: From the properties of polar form:
nhaa/imk/sem /eqt101/rk12/32

40. nhaa/imk/sem120162017/eqt101/rk12/32
De Moivre’s Theorem
From the properties of modulus: And suppose:
nhaa/imk/sem /eqt101/rk12/32

41. nhaa/imk/sem120162017/eqt101/rk12/32
De Moivre’s Theorem
Using all these facts; (3),(4) and (5), we can compute the square of a complex number. Suppose so Then
nhaa/imk/sem /eqt101/rk12/32

42. nhaa/imk/sem120162017/eqt101/rk12/32
De Moivre’s Theorem
Theorem 3: If is a complex number in polar form to any power of n, then De Moivre’s Theorem: Therefore :
nhaa/imk/sem /eqt101/rk12/32

43. nhaa/imk/sem120162017/eqt101/rk12/32
De Moivre’s Theorem
Example: a) Let Find b) Use De Moivre’s theorem to find: (i) (ii) Answer : a) b) (i)
nhaa/imk/sem /eqt101/rk12/32

44. De Moivre’s Theorem : Finding Roots
We know that argument of z is not unique, then we can also defined
Using the fact above and DMT, we can find the roots of a complex
number,

45. De Moivre’s Theorem : Finding Roots
Theorem 4 If then, the n root of z is: (θ in degrees) OR (θ in radians) Where k = 0,1,2,..n-1

46. De Moivre’s Theorem : Finding Roots
Example Find all complex cube roots of Solution: We are looking for complex numbers z with the property With

47. De Moivre’s Theorem : Finding Roots
k=0: k=1: k=2:

48. De Moivre’s Theorem : Finding Roots
Sketch on the complex plane: 1 y x
nth roots of unity:
Roots lie on the circle with radius 1

49. De Moivre’s Theorem : Finding Roots
Example: Solve and show the roots on the Argand diagram.

50. De Moivre’s Theorem : Finding Roots
Solution: Therefore, k=0:

51. De Moivre’s Theorem : Finding Roots
k=1: k=2: k=3:

52. De Moivre’s Theorem : Finding Roots
Sketch on complex plane: y x

53. nhaa/imk/sem120162017/eqt101/rk12/32
COMPLEX NUMBERS
Expansions for cosn and sinn in terms of Cosines and Sines of multiple n
Loci in the Complex Numbers
nhaa/imk/sem /eqt101/rk12/32

54. Expansion of Sin and Cosine
Theorem 5: If , then: Theorem 6: (Binomial Theorem) If , then:

55. Expansion of Sin and Cosine
Example Expand using binomial theorem, then write in standard form of complex number: Answer: a) b)

56. Expansion of Sin and Cosine
Example State in terms of cosines. Solution: By applying Theorem 5 and Binomial Theorem
Theorem 5

57. Expansion of Sin and Cosine
Solution: By applying Theorem 5 and Binomial Theorem
Theorem 5

58. Expansion of Sin and Cosine
Expand LHS using Binomial Theorem:

59. Expansion of Sin and Cosine
Expand RHS: Equate (2) and (3):

60. Expansion of Sin and Cosine
Example: By using De Moivre’s theorem and Binomial theorem, prove that:

61. Expansion of Sin and Cosine
Solution: By applying DMT: Expand LHS using Binomial theorem and let :

62. Expansion of Sin and Cosine
Equate (1) and (2), then compare: Real part: Imaginary part:

63. Expansion of Sin and Cosine
Example Using appropriate theorems, state the following in terms of sine and cosine of multiple angles : Answer: a) b)

64. Loci in the Complex Numbers
Since any complex number, z = x+iy correspond to point (x,y) in complex plane, there are many kinds of regions and geometric figures in this plane can be represented by complex equations or inequations. Definition 1.9 A locus in a complex plane is the set of points that have specified property. A locus in a complex plane could be a straight line, circle, ellipse and etc.

65. Loci in the Complex Numbers
(i) Standard form of circle equation Equation of circle with center at the origin, z0 , O(0,0) and z=x+iy, P(x,y) and radius, r y x
P on circumference:
P outside circle:
P inside circle:

66. Loci in the Complex Numbers
(i) Standard form of circle equation Equation of circle with center, z1= x1+ iy1, A(x1, y1) and z=x+iy, P(x,y) and radius, r

67. Loci in the Complex Numbers
Example: What is the equation of circle in complex plane with radius 2 and center at 1+i Solution: Re Im
Distance from center to any point P must be the same

68. Loci in the Complex Numbers
(ii) Perpendicular bisector where the distance from point z=x+iy, P(x,y) to z1 =x1+iy1 , A(x1,y1 ) and z2 = x2+iy2, B(x2,y2 ) are equal. Locus of z represented by perpendicular bisector of line segment joining the points to z1 and z2

69. Loci in the Complex Numbers
Example Find the equation of locus if:

70. Loci in the Complex Numbers
Solution: * Locus equation : A straight line equation with m = -2

71. Loci in the Complex NumbersRe Im
Distance from point (0,-1) and (2,0) to any point P must be the same

72. Loci in the Complex Numbers
(ii) Argument: The locus of z represented by a half line from the fixed point z1 =x1+iy1 , making an angle, with a line from the fixed point z1 which is parallel to x-axis

73. Loci in the Complex Numbers
Example If , determine the equation of loci and describe the locus of z

74. Loci in the Complex Numbers
Solution:

75. Loci in the Complex Numbers
Example Find the equation of locus if: a) b) c) Answer: